Normalized grounded states for a coupled nonlinear schr\"{o}dinger system on R³

We investigate the existence of normalized ground states to the system of coupled Schr\"odinger equations: {equation}{eq:0.1} {cases} -Δ u_1 + λ_1 u_1 = μ_1 |u_1|p_1-2u_1 + β r_1|u_1|r_1-2u_1|u_2|r_2 & in R³, -Δ u_2 + λ_2 u_2 = μ_2|u_2|p_2-2u_2 + β r_2|u_1|r_1|u_2|r_2-2u_2 & in R^3, {cases} {equation} subject to the constraints Sa₁ × Sa₂ = \(u₁ ∈ H¹(R³))|∫R³ u₁² dx = a₁²\ × \(u₂ ∈ H¹(R³))|∫R³ u₂² dx = a₂²\, where μ₁, μ₂ > 0, r₁, r₂ > 1, and β ≥ 0. Our focus is on the coupled mass super-critical case, specifically, 10/3 < p₁, p₂, r₁ + r₂ < 2^* = 6. We demonstrate that there exists a β̃ ≥ 0 such that equation ({eq:0.1}) admits positive, radially symmetric, normalized ground state solutions when β > β̃. Furthermore, this result can be generalized to systems with an arbitrary number of components, and the corresponding standing wave is orbitally unstable.